A common method to identify communities in a social network is to partition it in a way to maximize a quantity known as modularity. [1]} We shall use the notation \(C= \lbrace c_1,\ldots ,c_q\rbrace \)  to denote a partition over set \(V\)  where each \(c_i \in C\)  is a subset of \(V\) , for any \(c_i,c_j \in C\) , \(c_i \cap c_j = \emptyset \)  and \(\bigcup _i c_i = V\) . For a given partition, \(C\) , the modularity \(M(C)\)  is a number in \([-1,1]\)  . The modularity of a network partition can be used to measure the quality of its community structure. Originally introduced by Newman and Girvan [1]}, this metric measures the density of edges within partitions compared to the density of edges between partitions. A formal definition of this modularity for an undirected network is defined as follows.
